SOME PROBLEMS OF STATIC ANALYSIS OF FOLDED PLATE STRUCTURES

SOME PROBLEMS OF STATIC ANALYSIS OF FOLDED PLATE STRUCTURES

Lajos KOLL.A..R

Department of Building Construction Faculty of Civil Engineering Technical University of Budapest

H-1521 Budapest, Hungary Received: June 2, 1993

Folded plate structures are extensively dealt with in the literature, but there are some neglected probiems the solution of which the design engineer needs. In this paper are treated: the static behaviour of folded plate structures under partial vertical and un- der horizontal loads; static analysis of the extreme elements, of triangular plates loaded perpendicularly to their planes, and of plates subjected to in-plane concentrated forces;

stability analysis of the plate elements.

Keywords: folded plate structures, plates subjected to perpendicular loa.ds, plates sub- jected to in-plane loads, stability of plates.

1.

The classic analysis of folded plates can be found in the literature, see e.g.

(BORN, 1954, 1965). There are, however, some special problems which are not treated there, although the design engineer would need their solutions.

In this paper we want to deal with some of these problems.

First of all, we have to clarify some notions. We can distinguish basically two kinds of folded plate structures: those consisting of long and of short plate elements. An element is called long if its length is several times larger than its width (Fig. 1.1); it is called short if these two dimensions are close to each other (Fig. 1.2). The folded plate structures with long elements can be subdivided into barrel-vault-like (Fig. 1.la) and periodic ones (Fig. 1.lb).

We shall treat the following problems:

In the structures with long elements:

to what extent a partially loaded fold is supported by the unloaded neighbouring folds?

how the folds can be analysed when subjected to horizontal loads?

what are the internal forces in the extreme plate element due to con- centrated supports?

how the plate elements can be analysed for buckling?

168 L. KOLLAR

1.1 a

1.1 b

1 ^J.

Fig. 1.1.

In the structures with short elements:

how large are the bending and twisting moments in the triangular plates under a load perpendicular to their planes?

what stresses arise in the triangular or quadrangular element due to concentrated forces acting in the plane of the plate?

how the triangular plates can be analysed for buckling?

1.20. 1.2 b

1.2.

Before Gearing Vilith the nrot}le:ms of i::1Lli::11Ybl1:i, we have to survey the static

beha;viour of folded elements 1

2.1. Folded Plate StT'uctZtf'es with Elements

The element carries the to its of the

(in most cases distributed) load and transmits it to the edges forming the border of the element. Here the supporting forces have to be resolved into components acting in the planes of the adjacent two plates, which carry them by bending (causing tension and compression) and shear, both in their own planes, to their supports. The components of the load which lie in the planes of the plate elements also cause internal forces acting in the planes of the elements.

In this 'longitudinal load-bearing' the individual plates do not act alone, they rather form, together with the adjacent elements, great thin- walled beams which are the load-bearing elements proper in the longitudi- nal direction. This acting together is ensured by the so-called edge forces which are actually shearing forces transmitted from one piate to the other along the edges (Fig. 2.1), and whose distribution can be considered, on a structure simply supported at both ends, with close approximation as a r:osine function. Since these edge forces cause elongation also along the opposite edge of the element, we can ensure the identical elongation of the

170 L. KOLLAR

Fig. 2.1

common edge of two adjacent elements only if we consider, besides the edge force acting along the edge investigated, also the edge forces acting along the two neighbouring edges. thus obtain the so-called three-edge-forces equations, analogous to the three-moment equations of continuous beams.

The structures with long elements can be further characterized by the fact that the edges of the elements may considerably deflect due to bend-

in the plane of the element (above all if the length of the elements is manifold of their width), and these deflections interact with the primary bending of the plates (their load-bearing in the transverse direction), be- cause they appear as deflections of the supports of the continuous plate.

Hence a separate investigation is needed to decide when it is permissible to consider the edges as immovable with respect to the transverse bending of the plate.

If the deflections of the edges (i.e. of the suppol1;s of the plate) cannot be negh;cteQ with ref>Di(;ct to the transverse beJrl.Qllng of the then due to these deflections the forces of the plate change, which also modify the loading of the longitudinal beams. Hence, the deflections of the longitudinal beams and the transverse bending moments of the plate interact.

Summing up, the internal forces of the folded plate structure are de- scribed by a system of equations in which the edge forces and the trans- verse bending moments of the plate (at the supporting edges) appear as unknowns (BORN, 1954).

The static behaviour of the folded plate structure becomes so com- plicated, as a rule, only in the case of barrel-vault-like structures, because the long plate elements cannot form a (more or less) unique cross-section without edge forces. Furthermore, the planes of the adjacent plates mostly

subtend a small angle, so that they cannot hold the edge immovable. Hence, for simplifying the analysis, criteria have been established many years ago, whose fulfilment ensures that the edges can be considered as immovable, i.e. the plates can be analysed as continuous beams on fixed supports, and in the three-edge-forces equations only the edge forces appear as unknowns.

Thus, no intera.ction takes places between plate bending moments and edge forces.

The condition has been established (BORN,

ac,cOl~dinQ' to which the edges can be considered as immovable if the angle

two is at least 40° .

Fig. 2.2

More detailed conditions originate with Griining (BORN, 1954), which state that (Fig. 2.2b) if

t² 84

-2 :::; 1.6 sin 'P tan 'P-::::i"'

s L-

then the edges can be considered as immovable; and if

then the structure can be treated as an ordinary beam with a unique cross- section which does not deform, i.e. in which the distribution of the bending stresses is linear.

Finally, if t / s lies between the two above limits, the deflection of the edges has to be taken into account when computing the transverse bending moments of the plate.

In the latter case we may solve the problem by iteration: in the first step we determine the bending moments and support forces of the plate,

172 L. KOLLAR

assuming immovable edges; from these we compute the in-plane forces and edge forces of the plates and their edge deflections; then the influence of these deflections on the transverse bending moments of the plate and the pertinent support forces, etc. Since the transverse bending moments of the plate do not vary in the longitudinal direction (except for the vicinity of the end diaphragms), but the edge deflections do, for simplicity it is advisable to assume 2/3 of the maximum edge deflection as a constant value along the whole edge.

This iteration has not only the advantage to avoid the direct solu- tion of the large equation system, but it is also visual: we can follow the magnitudes of the different effects, and we can also see when the iteration procedure can be stopped. So in certain cases we may allow that the edge deflections cause nonnegligible bending moments in the plate, but we re-

that the support forces due to these moments only slightly alter the

forces~

VVe may also mention the so-called membrane of folded plates.

This stipulates edges, so that the plates carry the load as simply ones. Consequently, the edge forces do not interfere 'lNith the transverse of the plate, furthermore the load components acting in the planes of the plates can be determined from equilibrium equa- tions Hence we only have to solve the three-edge-forces equations.

The membrane is thus a simple, consistent of assumptioJD.s.

It has, however, a serious disadvantage: it does not reliably describe the static behaviour of any folded plate structure, since the r.c. plates are in every case continuous in the transverse direction. Hence this is suitable for theoretical investigations for this purpose it is, hc~wlev,er.

very useful, since it is the edge forces which ensure the connection between the plate elements in longitudinal In the frame of the membrane theory these edge forces can be determined by an iteration also al1,alog;otlS to the well-known moment distlritm1;ion method of frames. The 'effective width', the unloaded folds, will also be determined in Sect. 2.2 by using the membrane theory.

The static behaviour of the periodic folded plate structures (Fig. 1.lh) is much simpler than that of the barrel-vault-like ones, since in the main load-bearing (under a load uniformly distributed in the transverse direc- tion) no such edge forces arise which should be determined in the above described way, and their edges deflect identically. the structure with a triangular cross-section no edge forces develop at all, and in that with a trapezoidal cross-section the arising edge forces can be determined by the elementary strength of materials). This very simple state of stresses will be disturbed only by the fact that the structure is finite in the transverse di- rection: here the homogeneity of the structure ceases to exist, and the edge

forces caused by this fact should be computed as described above.

caily, however, we can content ourselves with guessing the internal (KOLLAR, 1993), These 'edge disturbances' rapidly die out with im:reamng distance from the edges, see in Sect. 2.2.

KOVACS (1978) investigated in detail the propagation of dis- turbances on the folded plate structure with triangular Cr4)SIiJ:-s~~ctlOI15 and presented the results in diagrams.

In the foregoing we assumed the folded

only bending in the longitudinal direction. There are, howeve:l\ struc-tmes which undergo bending and compression (Fig. 2.3). these cases aforementioned principles have to be the dilne:nsiOIlln.g

2.3 b

Fig. 2.3.

2.2. The 'Effective Width' Represented by the Unloaded Folds If only one fold will be filled by snow or industrial dust (Fig. iLIa), then the neighbouring, unloaded folds help the loaded one in carrying the load,

174 L. KOLLAR

mainly by the edge forces arising between them. Essentially the same phe- nomenon takes place if one (e.g. the extreme) fold is supported (Fig. 2.4b):

we then investigate how this disturbance propagates.

2.4 a

2.4 b

/~///

I

Fig. 2.4.

'"

/ "

/

,

,

,

..

"

.. ,

.. "

..

^"

In order to clarify this phenomenon we will study the magnitude of the edge forces on the unloaded part of the structure.

As mentioned earlier, the three-edge-forces equations are analogous to the three-moment equations of the continuous beams. Hence let us consider such a beam with equal spans on fixed supports, infinitely long in one direction, and let us i.nvestigate the magnitudes of the bending moments at the due to an external moment at the left support;

i.e. let us clarify how the influence of dies out.

Fig. 2.5.

a.

b.

~---2S-;;;--2---2S-X--3---""'LS4 ^{- - - -}

Fig. 2.6.

The three-moment equation of an unloaded beam runs (Fig. 2.5):

( 1' -)

Here the subscripts 1, m and r denote left, middle and right, respectively.

First let the bending stiffness El of the beam be constant, its spans be of equal length 1, see Fig. 2.6a. In this case the moment Ml decreases identically in each span to the a-fold of its value (lal

<

1). Thus the three-moment equation becomes:

(2a) i.e.

1

+

4a

+

a 2

=

^{0, (2b)}

which yields, taking into account the requirement that lal

<

1:

a

=

-0.27. (3)

116 L. KOLLAR

us now assume that only every second span together with the bending rigidity is identical (Fig. 2.6b). The decrease along the spans

1I

is then aI,

along the spans

h

it is a2. The three-moment equations can be written for the supports 1-2-3 and 2-3-4:

h ( h ·

12 ) 12

Ell Ml

+

2 Ell

+

El2 M2

+

Eh M3

=

^0,

12 11

1j

=

Eh : Ell maJWJig use of the fact that

1 ^I -'-I

COll§Hlermg the requirement

lail <

1:

(4a)

(4b)

(5)

(61'1)

(6b)

(6c)

(81'1)

(8b) If e.g. fJ = then al = -0.19, 012 = -0.34, their average value (or more exc!d1y, the value of .,jOll0l2 ) is thus very close to -0.27, see Eq. (3).

We can draw the conclusion that on continuous beams according to Figs. 2.5 and 2.6 the moment applied at the first support decreases at the third support to less than 10% of its original value.

Let us apply these results to the edge forces of the folded plate, using the membrane theory (i.e. assuming hinges along the edges).

The three-edge-forces equations of the unloaded folded have the following form (cf. 2.7):

1

+2

1 .I.

+

^=0.

Fig. 2.7.

Hence the quantity El/l characterizing the bending stiffness of the beam is substituted for by the area of the plate element, cf. also Eq. (1). Thus, on the folded plate with triangular cross-section according to Fig. 2.7a, the edge force decreases along one element to its -O.27-fold, while on the folded plate with trapezoidal cross-section (Fig. 2.7b) it decreases according to the expressions (8). It should be remarked that if we choose, on the latter type of structure, the thickness of the plates in such a way that the product ts be equal on both plates, then the role of the plates in decreasing the edge forces becomes identical, and these forces decrease to their -O.27-fold along each plate.

From all these follows that the influence of a partially loaded or a supported fold extends practically to two adjacent plate elements only.

Knowing the law of decrease of the edge forces we can determine an 'effective width' for the folded plate, which statically replaces the unloaded

178 ^L. KOLLAR

folds with respect to their supporting effect on the loaded one. By defini- tion, an edge force acting on the effective width causes a constant stress 0"1

in its whole cross-section. To ensure static equivalency it is thus necessary that this stress ^0"1 be equal to that caused by the edge force at its own point of application on the folded plate structure (Fig. 2.8a,b). Actually, the ef- fective width should be depicted as in Fig. 2.8c to show that a constant stress arises along its whole width.

c.

b.

I-

'I

\..-.

,\

unloaded part loaded fold

a

... -$E s"

::,

"2 _'\ l,. 2"

;(

1 b

s>;

"2

c

2.)\;

2 "~I

'"

Fig. 2.8.

k

unloaded Dart 1

d

Let us examine a fold consisting of two plates (Fig. 2.9a), as a cut-out part of a large structure, acted upon along one of its edges by an edge force

o. b. c.

1I1111111/1111~

1-o.27T

Fig. 2.9.

and let us determine the stress arising at the loaded edge. Since along the opposite edge of the loaded element the -0.27-fold of the edge force acts, the plate element is loaded by the two edge forces shown in Fig. 2.9b.

These cause at the loaded left edge a stress

T

+

O.27T (T - 0.27T) ~ T

0"1

=

_ts

+

₍ ^t _~-) ^?

=

^3.46-, _ts ⁽¹⁰⁾

while at the opposite edge the -0.27-fold of this value arises.

If we want that the uniform stress T / (ts"') inside the effective width s* be equal to 0"1, the equation

~ =

^3,46T

ts" ts ⁽¹¹⁾

must hold, which yields for the effective width the value

s*

=

^{0.29s, (12)}

see Fig. 2.9c. Substituting this effective width for the unloaded folds ac- cording to Fig. 2.8b, we obtain a visual picture of their supporting effect.

180

Cl.

\ I

\....-1

b.

I I

\ \

wind force

\

p - -

1

L. KOLLAR

dust load

\ I

\ I

'---'

2. 1fJ

r~ 7'", .~;r\",

1/

h

Fig. 2. 11

2.3.Analysis of the Folds for Horizontal Loads

In folded plate structures containing steep plate elements the load may bend the structure in the horizontal plane (Fig. 2.10a). The wind causes a similar effect if the folds horizontally support the side

wan

(Fig. 2.1(0). The structure then acts essentially in the same way as under vertical load: the plate elements carry the load as large beams on the span.

L. If the horizontal deflection of the edges cannot be neglected, the elements will act as transverse frames and transmit the horizontal load the other folds. In (KOLLAR, 1974) we find an approximate method determine the internal forces due to this of load, the as:o;mnp,ti()llS which are the fol.101wiJl£f:

the one the inclined elements of the folded

can be considered steep enough to neglect the vertical deflections of th~ll

edges; on the other hand, the horizontal elements are narrow en,ongh prevent the rotation of the edges of the inclined elements. Hence the struc- ture deforms as shown in Fig. 2.11: the horizontal with a section (effective 'mdth) of the inclined ones constitute U-shaped beams which are connected by springs formed by the bending stiffness of the inclined 1.l'Ji.(:~"ee;.

So the structure can be modelled as depicted in Fig. 2.12: the hori2;onltally bent beams are connected by distributed springs.

The method of analysis can be shortly summarised as follows:

The horizontal load p, uniformly distributed along the span is expanded into a Fourier series, of which we consider - for the time

only the first term (n

=

^1):

where the load amplitude PI is given by the expression PI

=

^-po 4

r.

The 'spring constant' c of the inclined plates, referred to unit length, is equal to the stiffness against displacement of an inclined beam built-in at both ends:

where

12Elpiate c= h²s

Elpiate

=

^{12(1 _ v}2 ) ,

see Fig. 2.11, and v is Poisson's ratio.

(14a)

(14b)

182 L. KOLLAR

p

=

p . sin It·X

1 L

<:1lf]tlltt~

J

^{L ' I}

^l

^{beams springs}

k

=

¹

)t,::: i 2

2 3

3 m-i m

Fig. 2. 12

The effective width of the inclined elements can be assumed, taking into account what has been said in Sect. 2.2, approximately as 0.25 times the plate -width B. bending stiffness in the horizontal of the 'beams' with U shaped cross-section will be denoted

The force ^rk behveen the bea..rns k and k

+

1 is given the difference of deflections r~ of the two bea..'11s. Olni·tting the multiplicator

we obtain

=

On the other the deflection of a beam is caused the difference of forces. the pa.ra.m1et"r

(16) characterising the stiffness of a beam, the deflections of the beams k and k

+

1 become:

Tk-l - Tk

(l7a)

Vk

=

₁₅

1

and

Vk+l

=

^{Tk -}₁₅^{Tk+l (17b)}

1

Introducing (17a,b) into (15), we arrive at the following homogeneous linear difference equation for the spring force r:

8-

rk-l - (2

+ -=-

_c ^)Tk

+

rHl

=

^O.

Assuming the solution in the form

rk

=

^{cp ,} k

and introducing it into (18), we obtain the characteristic eqll1al;lo:n:

+ +

=

whose roots are:

Pl,2

=

P c

81 ) ^I

+ -

::r:

, 2c Thus, the solution has the form:

This has to satisfy the boundary conditions:

k

=

^{0: TO}

=

^PI,

k=m: ^Tm

=

^O.

(18)

(1gb)

(20)

From these we determine the integration constants Cl and Cz, and the solution (20) takes the final form:

(21)

yielding the maximum value of the spring force k, varying according to a sine curve along the span L.

The beams are loaded by the differences of the spring forces acting at their both sides. It is the first beam which carries the greatest share of the load

(22) so that the total load PI acting on the first beam decreases to (PI - rI).

184 L. KOLLAR

All what has been said so far refers to a load varying according to a half sine wave, see Eq. (13a). Since the Fourier series of the uniform load

p= -p 4

1i n=1,3,5 ...

sin ( n1ix/

L)

n ⁽²³⁾

the load corresponding to the n-th Founer term has the form pn = - -4p

1in (24)

and thus

p=

E

pn sin(n1ix/ L). (25)

n=1,3,5 ...

ll'C.!LIf.;':;, in the higher terms (n

=

^{3,5 ... )} we have to replace L by L/n,

and we ha.ve to compute the pertaining bn according to (16). The bending moment of the bea.m is obtained by differentiating (25) twice:

M=

~2 E

" n=1,3,5 ...

sin(n1ix/ L). (26)

Due the action of the springs, for the first beam we have to substitute a{;cording to (22), and for the other beams we have to instead of p, every r with the n in question.

of the moment ahovls that

Element n/;stzng

decrease to zero the 110rizontal dimensioE x of e::;:treme element reac!::!.llll.g bc,y()n,d the first SUlP:pc,rt then the first

which would continuous 81.JljJ1DCil't of the second

el~err!.ellt, ceases to e;;dst, so that this second

moments can

taken from (STIGLAT and 1983), in of

the moment distribution of a cam'GUev·er

which the diagra.ms infinitely wide ^III found (Fig.

one direction and loaded at its corner point, ca.n be

Q

b

thus obtain the moments of the extreme in the

the

V'lay: In the first we assume a continuous StLpp,ort edge at the first

In the second continuous stlPP'Drt two corner points.

and determine the moments due to the load.

we add to these the moments caused the concentrated

vvhile the

those VVe can see that the

crest theSe a zero sum

t=-ans"tverse moments decrea.se tC}VJ'a:rd.8

rules in 2.6.

2.5. Problems

structures have two kinds of stabil- elements (loca.l buckling),

The intermediate

1. • 4 1 .1. and. '6}Q) ^4. <Y •

supports along all edges. In b,':n.d.Jmg according to

The compression and/m: bend-

but due to their free edge their critical load intensity is lower than of the intermediate ones (Fig. 2.16c,d,e). The numerical values have

186 L. KOLLAR

a. m y in sec,:on y=O

-1. 55

my

=

^kP

-0,77

! -0.57 I

· _ -0.43

-0.19

-+-_-'---'-I_-'-_-'-:---'_-'-_...:..! :--""'..2L

10 0.2 0.4 0.6 0.8 1.2 1.4 Ly

b.

kA

1-1.55

l\ I \~0.B2

I 1~·52

11

! i '-1,3.\

_111~

o ^{0.2 0.4 0.6 0.8 1.0}

section x=O

m,

^{= kP}

c.

10 0,2 0,4 016 O/~ 1.~ 12 ^{' !} if"";'

d. my= 0 in secr;or

e. l; m, in section y=Ly ^{m ~ =kP}

-0.15 -0.20 -0.20 -0.19 -0.16 _-0.14

I I i i ^-0.11 L

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ly

f. m x = 0 in section X=O Fig. 2.

14

o.

b.

/

Fig. 2. 15 I /

/

/ /

188 L. KOLLAR

o. b.

c. d.

n1 n1 (-nl) (-;-, 1)

8~ !~ ^AI

^nl

^YII

^r

^~\

ⁿ¹

kzO,608(l-i/) k.l,Z16 (1-\))

[Timoshenko and Ger!?, 19611 [Vo.jnberg, 19731

e.

r.l

n,

\11

(-rl1)

~l

(-nil

i Q ~{ate thc:-<:ness

V ^.= Poisson s ;u~io

= ~ hinged Edge

- G free Edge

kcZl, ^0. » ^b

N=O) (Ambrcizy, 1984 J

Fig. 2. 16

II!!11IIII11111II!

Cl.

2.17

Fig. 2. 18

been taken from (TIMOSHENKO and GERE, 1961), (VAJNBERG, 1967) and (AMBROZY, 1984).

All there results refer to plates under bending or compression con- stant along their length. However, folded plate structures undergo, as a rule, a (parabolically) variable bending moment. Thus if we consider the maximum bending moment as constant along the entire length, we commit an error to the benefit of safety.

190 L. KOLLAR

'" /v~

"-

'"

","'-_. ^{' .} . -~"

"

" "

b b

1

Fig. 2. 19

If the plate element undergoes at the same time bending and com- pression, we ca..l1 use the Dunkerley formula, see e.g. in (KOLLAR, 1991),

result of lies on the safe side:

faz:tor.

oucKlIng the structure has an ca-

in the elastic range, see e.g. in (KOLLAR, 1991). the plasticizing of the material and the of the concrete cause a reduc- tion of the stiffnesses, resulting sooner or later in a decreasing load- bearing capacity.

Overall buckling of the whole structure occurs if its cross-sections deform (flatten). This comes about first of all if no diaphragms (ties) are provided at the supports. In the case of barre1-vault-like structures this generally does not happen, since diaphragms are needed to keep the shape of the cross-section anyway. Periodic folded plate structures, how- ever, can be built without diaphragms (or ties). The folds can thus 'slide apart' (Fig. 2.17), the structure flattens due to bending and snaps through

(Fig. 2.18). The critical bending moment causing snapping has been cal- culated by KEREK (1981), assuming that the cross-sections are 'flat'. We present the value of the critical bending moment referred to the width of one fold, for V and sine wave formed cross-sections and constant wall thickness, in Fig. 2.19. The numerical values of the sine wave formed cross-section can be considered as valid for the trapezoidal cross-section as well.

After snapping the structure has a decreasing post-buckling load- bearing capadty even in the elastic range, see (KOLLAR, 1973, 1991).

sequently, we should choose a higher safety factor than for plate with increasing load-bearing capacity.

We have to mention two further modes of buckling, the critical loads of which have not been determined. The first is the of the structure by as described but "'r1th not sild.Ing apart (Fig. 2.20). This may occur only in structures with very

elements, where the middle part of the structure slides and fiattens.

This deformation is made possible the shearing deformation of the elements in their own planes.

2.20

The second mode of instability is the buckling in the horizontal plane of the upper, U -shaped part of the trapezoidal folded plate structure (Fig. 2.21). This is possible only if the vndth of the 'U sections' is small, and the inclined plate elements provide only a weak lateral support.

Fig. 2. 21

: I

\

l

192 L. KOLLAR

All we can say about these two buckling modes is that they may occur only in structures of extreme geometry.

3. Problems of Analysis of Folded Plate Structures Consisting of Short Elements

3.1. BehavioU1' of Folded Plate Structures with Short Elements

Folded structures consisting of short elements (Fig. 1.2) differ from elements in the following:

elements are 'short', and mostly triangular;

due to the shortness of the elements, the forces do not cause elongation in their opposite edge (since in a deep 'wali-beam' the stresses of the loaded edge die out untii the opposite edge);

the elements are each and not

fixed SUP1)Ort13. Due to the shortness of the elements their can considered as unmovable with to the bending of

The tnree-'~d:ge·-lClrces eQucLtlon.S what has been said of the Cl/fall-beams'.

most elements are supj:)OJrtc:d each

"1l1hose bars are formed load as a space

accord.mg to due to the fa"ct that the structure carries the

the &5 is shown

el,:IDLerlts does not esseJlltiaiJly

transmit also here the cc,m.p()nent

the b.:m(lil1g to the where these must be resolved into com- in the planes of the two joining plates. These components, to'g.::ttler with the load cause in-plane bending and shear in the plates. The deviation from the structures vlith long elements

IS that the plates subjected to in-plane forces are supported only the since it is here where several edges meet, which can support the vertex. Thus a 'space grid' is formed by the edges (together with some 'effective width' of the plates) as bars (Fig. 3.1), and this grid carries the load to the supports. The analysis of space grids can be found in (KOLLAR and HEGEDUS, 1985).

3.1

The Static Behaviour the

For the Cl:naIVS1S of the folded structures 1itlith short elements Vie have to know the moments and support forces of the tr:ial1gul,u

In (BREITSCHUH, diagrams can be found for these quantities.

The forces along the edges of the elements of ", .. h,t.r",..,r

shape can also be determined in the following simple, way.

c.. b.

Fig. 3.2

194 L. KOLLAR

Starting from the corners we draw the bisectrix lines and, if they do not in- tersect in one single point, we connect the intersection points (Fig. 3.2a,b).

The geometrical figures thus obtained yield the forces acting on the sides, together with their distribution.

The forces transmitted from the triangular plates to the vertices, nec- essary to know the internal forces of the whole structure (as a space grid), can also be simply determined. These forces have to be statically equivalent to the load acting on the plate. The problem can be solved by considering the plate as a 'table with three legs', which can be easily solved by using equilibrium equations only (Fig. 3.3).

Equiti brium equations:

3 ^{.<J Q}

I.MAl3 =0

!MOC ::0 uAAi:; =0 (LP ::;0)

As far as the forces loading the plate in its own plane are concerned, we find that consist of three parts: the components of the load lying in the plane of the plate, the support forces resolved in the planes of the two plates, and the bar forces of the space grid, which act from the vertices along the edges. The latter ones constitute a force system being in equilibrium and tJ:le former t'rVO force groups are held in the

lIl--Pli::Hl.e COIlJ.porleIlts of the support forces at the vertices.

Applying these force systems (all being in equilibrium) to the plate element, "ve can calculate the in-plane forces in the plate. However, we have to make one remark.

Since the 'bar forces' of the space grid act along the edges, we have to divide them between the effective widths of the two adjacent plates. The basic principle of the division should be the compatibility of deformations:

the two adjacent plates should undergo the same compression (elongation) due to the forces acting on them. For simplicity, however, we may divide these bar forces in half and half, provided the widths and thicknesses of the two adjacent plates do not differ considerably. The 'effective width' of the plate, valid for this case, will be treated in the next section.

3.3. The Stresses Arising in the Plate due to Concentrated In-plane Forces

In folded piate structures with short elements, concentrated forces act on the plates along their edges. These forces are mostly applied at the corner points of the elements which behave as 'discs' (i.e. plates loaded in their own planes). Such forces act along the inclined edges of the structure shown in Fig. and a horizontal tension force acts e.g. the edge of the element Along the inclined and horizontal edges of the structure of 3.5 also act concentrated forces, and vertical forces at A and B of the element ABeD. The 'bar forces'

of the structure of

o

') I '-'-"t-

.2b have a similar effect,

The is: what stresses are caused inside the disc these concentrated forces. To put it in another way: what is the 'effective YV1dth^j of the disc subjected to a concentrated force acting along its edges?

The problem can be solved by the theory of elasticity. Before pre- senting the results, however, we shall try to give a clear picture of the phenomenon by simple reasoning. (When in the following we shall speak of 'stresses', we actually mean 'stresses multiplied by the thickness of the disc': n

=

^to').

Let us consider the rectangular disc subjected to two concentrated forces along one edge (Fig. 3.6). The vertical sections of the disc un- dergo eccentric compression. The 'natural' state of stress of an eccentrically compressed bar is the linear stress distribution, given by the elementary strength of materials (Fig. 3.7), since the pertaining strain energy becomes a minimum as compared with other (curvilinear) stress distributions. This linear stress distribution, however, can come about only at a certain dis- tance from the cross-section where the force acts, since in this cross-section

t~e stress distribution corresponds to the application of the concentrated force (Fig. 3.8). According to the principle of Saint-Venant, this distance

196

p

/ /

L. KOLLAR

/

/

/

o

C!lb· / ~==~=='

3.5.

is to the height of the cross-section, because the of the con- centrated force can be composed from two parts, see Fig. 3.9: load case a) causes a linear stress distribution in every cross-section of the disc, while load case b) being a force system in equilibrium - causes stresses only inside a length which is equal to the loaded section, i.e. H.

. 1

I . ^~H

P P -L ^I

\

L

~

Fig. 3.6.

It follows from the foregoing that if the height of the disc is not greater than half its length (H

S

L/2), then the stress distribution in the middle

rly ::: ^eo

b...~~~~_~==

Fig. S.8.

cross-section is linear, and the resultant RI of the compressive stresses acts at a distance (2/9}H from the compressed edge.

If, on the other hand, the height of the disc is greater than its length (H

>

L), we can apply Saint-Venant's principle in the direction of the height: the force system in equilibrium causing disturbance consists of the two concentrated forces P, acting on the bottom 'cross-section', which cause stresses only inside a distance equal to L. Hence in the part of the disc above the height H

=

^L no stresses arise.

The state of stress of a disc with the height H = L, subjected to two concentrated forces, has been determined by BAY (1938) with the method of finite differences, subdividing the disc into 5+5 parts. His results are shown in Fig. 3.10a. The stress distribution of the section A - A can also be considered valid for the middle cross-section k - k, so the magnitudes

198 L. KOLLAR

+

-

p

G.,

Fig. 3.9.

o. b.

T T,-O"l

J

^I-OBL.H

compressed zone

,i.

^{I , '}

i I

?1~ 1,20P:z - - ; - -,- ;

! i-O 14H I

." _ _ ----.J r ^{I _ _ _} L

H=L

1⁷ig. 3. 10.

and situations of the resultants of the tensile and compressive stresses are given by Fig. :i. lOb.

It is worth while to note that the compressive stresses propagate into the inside of the disc according to the semicircle of the radius r

=

H /2.

The resultant Rl of the compressive stresses lies closer to the loaded edge than in the case of a linear stress distribution (Fig. 3.1).

3. 11,

For the disc loaded at its two upper corner (Fig. 3.11) we can a similar the effect of the t-wo forces can be replaced by a force system to the 'naturai' one ( a) and another being in equilibrium (b). The latter one causes stresses only inside a distance so that at this distance the distribution of the stresses becomes uniform.

p

K

p

Fig. 3. 12.

-I

K \

H=L

__ t_

I

If, on the other hand, a square disc is loaded by four concentrated forces (Fig. 3.12), then the stress distribution cannot become uniform until the middle cross-section k-k, but remains curved. This distribution is obtained by adding the diagram of Fig. 3.10a to its mirror image.

The internal forces of the triangular disc (Fig. 3.13a) are, according to (BAY, 1938), similar to those of the rectangular one, with the difference that the stress distribution in the middle cross-section can be considered linear only if tan a: ~ 0.5, i.e. a: ~ 26.5°, which means that the height H of the triangle is not more than one fourth of the basis length L.

200 L. KOLLAR

a

P.~~

b

I

^L

I

y H=Ll2

p

r - - - L - - - 1

Fig. 3. 13.

The stress distribution the middle cross-section of a triangular disc with H

=

L/2 is shown in Fig. 3.13b. The magnitudes and situations ofthe result ants of the tensile and compressive stresses correspond approximately to Fig. 3.10b.

Stao:ilitu Problems

The constituting the folded

st.ructures with short elements are far not so well clarified as those rectangular plates. We found the solution of two cases in the literature: for the equilaterai and for the rectangular isosceles triangular plate (Fig. 3.14), see in (TIMOSHENKO and GERE, 1961) and (\VITTRICK, 1954). In the figure we give the critical compressive force of the hinged plates for hydrostatic compression (nx

=

^ny

=

^n).

For practical applications it is advantageous to replace the triangular plate by a rectangular one exhibiting an equivalent stability behaviour. For this reason we investigated the follovving problem: if we want to replace the isosceles triangular plate by a rectangular one having the basis of the triangle as one side (Fig. 3.15), what height the rectangular plate should have in order to yield the same critical load for hydrostatic compression as the triangular one?

a..

o.

r-I:~~.,

J

I a a ^I

I :

I I

a

s:.t1

^to _o ^{to L. .. '')}

r

3.

n

Fig, 3. 15.

b.

:= plate thickness

V:

Poisson' s ratio : hinged edge

Assuming hinged edges for both plates we found that for the two cases shown in Fig. 3.15a,b the height should be O,555h and O.667h respectively.

For flatter triangles we did not find any result, but considering the tendency of these two cases we may assume that for flatter triangles the height of the rectangle should be somewhat greater than O.667h.

All what has been said so far is valid for hydrostatically compressed plates. We could not set up a comparison for other loading cases. Hence

WE>, recommend to assume a higher safety factor when substituting a rect-

angular plate for the triangular one. The plates exhibit an increasing post-

202 L. KOLLAR

buckling load-bearing capacity, see Sect. 2.5, so the reasonings presented there should be considered.

The overall instability of the folded plate structures with short ele- ments is the buckling of the space grid consisting of bars along the edges.

This phenomenon can be investigated by a programme based on the second- order theory, i.e. which analyses the internal forces on the deformed shape.

Acknowledgement

The help of the OTKA (Hungarian National Science Foundation) Grant No. 684 is grate- fully acknowledged.

References

AMBROZY, Gy. (1984): Buckling of a Plate with one Free Edge, Subjected to Linearly Varying Compressive Stresses. Thesis (in Hungarian). Technical University, Budapest.

BAY, H. (1938): Uber einige Fragen der Spannungsverteilung in Dreieck- und Rechteck- scheiben. Der Bauingenieur, Band 19, S. 349-356.

BEcKER, S. (1966): Berechnung prismatischer Faltwerke nach dem Spannungsverteilungs- verfahren. Az EpitOipari es Kozlekedesi Muszaki Egyetem Tudomdnyos Kozleme- nyei. Budapest. Vol.XII. No.3. pp.5-25.

BECKER, S. (1969): Berechnung prismatischer Faltwerke mit Querschottversteifung. Pe- riodica Polytechnica SeT. Architecture. Vol. 13. No.3-4. pp.121-129. Budapest.

BORN, .]. (1954) Faltwerke; ihre Theorie und Berechnung. K.Wittwer, Stuttgart.

BORN, J. (1965): Faltwerke. Beton-Kalender, Teil n. S. 385-455. W.Ernst u. Sohn, Berlin.

BREITSCHUH, K. (1974): Dreieckplatten. Triangular Plates and Slabs. Berlin, W.Ernst u.

Sohn. (Bauingenieur-Praxis, Heft 9.)

KEREK, A. (1981): Static and Stability Investigation of Bent Folded Plates. Acta Techn.

A.cad. Sci. Hung. Vo!. 93. pp. 39-6.5.

KOLL..\R, L. (1973): Statik und Stabilitat der Schalenbogen und Schalenbalken. W.Ernst u. Sohn, Berlin Aka<:iemiai Kiad6, Budapest.

KOLLAR, L. (1974): A Simple Analysis of Folded Plate Roofs for Horizontal and Partial Loads and its Application to an Erected Structure. Proc. lASS Symp. on Folded Plates and Spatial Panel Structures. Udine (Italy).

KOLL..\R, 1. (1991): Special Problems of Structurai Stability Akademiai Kiad6, Budapest.

KOLLAR, L. (1993): Design of Engineering Structures (in Hungarian). In preparation.

KOLL . .\R, L. HEGEDUS, 1. (1985): Analysis and Design of Space Frames by the Con- tinuum Method. Elsevier Science Publishers, Amsterdam - Akademiai Kiad6, Bu- dapest.

Kov . .\cs, B. (1978): Internal Forces of a Periodic Folded Plate Roof. Doctor's Thesis.

Technical University, Budapest.

STIGLAT, K. - WIPPEL, H. (1983): Platten. 3 Aufi. W.Ernst u. Sohn, Berlin/Miinchen.

TiMOSHENKO, S. - GERE, J. (1961): Theory of Elastic Stability. McGraw-HiIl, New York.

VAJNBERG, D.V. (1973): Spravochnik po prochnosti, ustoychivosti i kolebaniyam plasztin.

Izd. Budivel'nik, Kiev.

W!TTRICK, W.H.(1954): Symmetrical Buckling of Right-Angled Isosceles Triangular Plates. Aeronautical Quarterly, Vol. .5, pp. 131-143.